In Unit 11, before you begin this unit you must have previous knowledge of what a GCF is, how to distribute, the process of answering exponential numbers/coefficients, how to apply and do the process of the systems of equations. Once you reach the end of this unit you will be able to factor, create a generic rectangle, know what a zero property product is and also know how to find a square roots as well as radicals. However, along the line of some key misconceptions that you can find in quadratics that would be incorrect placing negative and positive signs. Some type of challenge or word problems that would exist for this unit would be asking to calculate the total amount of apples brought from the market if in a group by color and multiply the group apples by size then to figure out how many apples are are in one group of color and the other in size.
In a Geo./Alg. 2:Derivation of Quadratic Formula: x2 + 3x – 4 = 0
Steps to solve this problem: First: The first step to this problem is figuring out what you are going to factor. Once that is done you should set it up behind the original problem in it’s parentheses sets and set everything equal to zero.
1.) x2 + 3x – 4 = (x+4) (x-1)= 0
Secondly, Now that the whole equation has been set up and equal to zero you have to use the Formula to find the solution. This formula looks like:
x = [ -b ± sqrt(b^2 - 4ac) ] / 2a
The “2” that is placed as the denominator is underneath everything, not just the square root. * After the formula is set up with the proper variables filled in you can begin to solve for the solution.
=-3+9+16 = -3+25 ----------- --------------- 2 (Divide on both sides) 2 = -3+5 = -3-3 , -3+5 -------- ------ --------- 2 2 2 ; (Divide by 2 under all sides of the equation) = -8 , 2 = -4, 1 = x --- --- 2 2 (Divide by 2 on both sides)
Finally, in conclusion the solution is x= -4 & x=1
An exact example from the SAT’s: a^2-b^2=12 and a+b=4 What is the value of a-b?
Steps to solve this problem: First: The first step to solving this problem is to know that both your a^2 and -b^2 will be factored. They will turn into two different things, one positive and one negative. This is because the subtraction sign in between them now makes one of the factored parentheses negative and the other positive.
1.) (a+b) (a-b)=12
Secondly: We know from previous information that there are technically two equations given in the example. The second one is “a+b=4”. This indicates that the one set of parentheses with “A” being added into “B” is equal to four. Therefore, we can now move a new variable into the position underneath the set of parentheses. Now you have to divide by 4 on both sides of the equal sign to find out what (a-b) are equal to.
1.) 4(a-b)=12 4 4 (Divide on both sides of the equation) [ a-b = 3 ]
Next: After you have found the set number for the negative parentists set you have to find the opposite. To do this you must do inverse operations. But first, you have to set up the positive parenthesis set equal to 4. 1.) a+b=4 -b -b a= 4-b
Once you have done that you can substitute your answer into the original problem and will get the correct answer. The man from my source where I obtained this example said it will take time though.
In a Real-Life situation: An object is thrown downward with an initial velocity of 19 feet per second. The distance, d it travels in an amount of time , t is given by the equation d=19t + 15t^2. How long does it take the object to fall 50 feet?
Steps to solve this problem: As you begin this example problem you have to make sure you understand what they are asking and how it can be done. Mainly any quadratic expression can be solved through the quadratic formula. In the equation give we can only have one unknown so we have to fill in the rest of the variables which we do know from the information in the text. 1.) d= 50 ft. After you found the unnecessary unknown which was really known you can begin to solve the quadratic using the quadratic formula. However, the equation has to be set to 0 or in other words, standard format. Standard Form: ax^2+bx+c=0 Since it all equations using quadratics have to be set to ceo we must do inverse operations and set our equation to zero. 1.) 15t^2+19t=50 -50 -50 15t^2+19t-50= 0
I apologize because the quadratic formula and large square root sign wouldn't upload correctly to the Wiki and I wanted to show all my work.
It would take around 1.3 seconds for it/an object to fall 50 feet. Lastly, you can check your work.
Created Problems: A.) x^2+17x+99
Steps to solve this problem: First: The first step you can take to solve this problem is by looking at what you are going to factor. Those numbers would be: 17 and 99. The number 17 has to have the same two numbers as 99 does but it has to add up to 17. In contrast, to the number 99 it has to be the same two numbers from 17 but multiply to 99. Secondly: set up your parentheses to place both the numbers in that are factoring both 17 and 99. Then decide if they are both or only one is negative or positive. Once you have done that then set up both x’s that are distributed from your x^2 in each parenthesis. 1.) ( x + ) ( x + )
Next: After you have finished that then you should find out the numbers and place them in the correct designated set of parenthesis.
1.) ( x + 11) ( x + 9)
Finally: after you have gotten both the number that add up and multiply into the correct number you are done solving the problem.
B.) x^2-8+15 Steps to solve this problem: First: The first step you can take to solve this problem is by looking at what you are going to factor. Those numbers would be: -8 and 15. The number -8 has to have the same two numbers as 15 does but it has to add up to -8. In contrast, to the number 15 it has to be the same two numbers from -8 but multiply to 15. Secondly: set up your parentheses to place both the numbers in that are factoring both -8 and 15. Then decide if they are both or only one is negative or positive. In this case they both are negative. Once you have done that then set up both x’s that are distributed from your x^2 in each parenthesis.
1.) ( x - ) ( x - )
Next: After you have finished that then you should find out the numbers and place them in the correct designated set of parenthesis. 1.) ( x -3 ) ( x -5 )
Finally: after you have gotten both the number that add up and multiply into the correct number you are done solving the problem.
In Unit 11, before you begin this unit you must have previous knowledge of what a GCF is, how to distribute, the process of answering exponential numbers/coefficients, how to apply and do the process of the systems of equations. Once you reach the end of this unit you will be able to factor, create a generic rectangle, know what a zero property product is and also know how to find a square roots as well as radicals. However, along the line of some key misconceptions that you can find in quadratics that would be incorrect placing negative and positive signs. Some type of challenge or word problems that would exist for this unit would be asking to calculate the total amount of apples brought from the market if in a group by color and multiply the group apples by size then to figure out how many apples are are in one group of color and the other in size.
In a Geo./Alg. 2: Derivation of Quadratic Formula:
x2 + 3x – 4 = 0
Steps to solve this problem:
First: The first step to this problem is figuring out what you are going to factor. Once that is done you should set it up behind the original problem in it’s parentheses sets and set everything equal to zero.
1.) x2 + 3x – 4 = (x+4) (x-1)= 0
Secondly, Now that the whole equation has been set up and equal to zero you have to use the Formula to find the solution. This formula looks like:
After the formula is set up with the proper variables filled in you can begin to solve for the solution.
x= -(3)+(3)2 -4(1)(-4)
---------------------------------
2(1)
=-3+9+16 = -3+25
----------- ---------------
2 (Divide on both sides) 2
= -3+5 = -3-3 , -3+5
-------- ------ ---------
2 2 2 ; (Divide by 2 under all sides of the equation)
= -8 , 2 = -4, 1 = x
--- ---
2 2 (Divide by 2 on both sides)
An exact example from the SAT’s:
a^2-b^2=12 and a+b=4
What is the value of a-b?
Steps to solve this problem:
First: The first step to solving this problem is to know that both your a^2 and -b^2 will be factored. They will turn into two different things, one positive and one negative. This is because the subtraction sign in between them now makes one of the factored parentheses negative and the other positive.
1.) (a+b) (a-b)=12
Secondly: We know from previous information that there are technically two equations given in the example. The second one is “a+b=4”. This indicates that the one set of parentheses with “A” being added into “B” is equal to four. Therefore, we can now move a new variable into the position underneath the set of parentheses. Now you have to divide by 4 on both sides of the equal sign to find out what (a-b) are equal to.
1.)
4(a-b)=12
4 4 (Divide on both sides of the equation)
[ a-b = 3 ]
Next: After you have found the set number for the negative parentists set you have to find the opposite. To do this you must do inverse operations. But first, you have to set up the positive parenthesis set equal to 4.
1.)
a+b=4
-b -b
a= 4-b
Once you have done that you can substitute your answer into the original problem and will get the correct answer. The man from my source where I obtained this example said it will take time though.
In a Real-Life situation:
An object is thrown downward with an initial velocity of 19 feet per second. The distance, d it travels in an amount of time , t is given by the equation d=19t + 15t^2. How long does it take the object to fall 50 feet?
Steps to solve this problem:
As you begin this example problem you have to make sure you understand what they are asking and how it can be done. Mainly any quadratic expression can be solved through the quadratic formula. In the equation give we can only have one unknown so we have to fill in the rest of the variables which we do know from the information in the text.
1.) d= 50 ft.
After you found the unnecessary unknown which was really known you can begin to solve the quadratic using the quadratic formula. However, the equation has to be set to 0 or in other words, standard format.
Standard Form: ax^2+bx+c=0
Since it all equations using quadratics have to be set to ceo we must do inverse operations and set our equation to zero.
1.) 15t^2+19t=50
-50 -50
15t^2+19t-50= 0
It would take around 1.3 seconds for it/an object to fall 50 feet.
Lastly, you can check your work.
Created Problems:
A.) x^2+17x+99
Steps to solve this problem:
First: The first step you can take to solve this problem is by looking at what you are going to factor. Those numbers would be: 17 and 99. The number 17 has to have the same two numbers as 99 does but it has to add up to 17. In contrast, to the number 99 it has to be the same two numbers from 17 but multiply to 99.
Secondly: set up your parentheses to place both the numbers in that are factoring both 17 and 99. Then decide if they are both or only one is negative or positive. Once you have done that then set up both x’s that are distributed from your x^2 in each parenthesis.
1.) ( x + ) ( x + )
Next: After you have finished that then you should find out the numbers and place them in the correct designated set of parenthesis.
1.) ( x + 11) ( x + 9)
Finally: after you have gotten both the number that add up and multiply into the correct number you are done solving the problem.
B.) x^2-8+15
Steps to solve this problem:
First: The first step you can take to solve this problem is by looking at what you are going to factor. Those numbers would be: -8 and 15. The number -8 has to have the same two numbers as 15 does but it has to add up to -8. In contrast, to the number 15 it has to be the same two numbers from -8 but multiply to 15.
Secondly: set up your parentheses to place both the numbers in that are factoring both -8 and 15. Then decide if they are both or only one is negative or positive. In this case they both are negative. Once you have done that then set up both x’s that are distributed from your x^2 in each parenthesis.
1.) ( x - ) ( x - )
Next: After you have finished that then you should find out the numbers and place them in the correct designated set of parenthesis.
1.) ( x -3 ) ( x -5 )
Finally: after you have gotten both the number that add up and multiply into the correct number you are done solving the problem.